How long to wait after configuring adc?

I'm multiplexing a processor pin between a digital-out (SPI/Clock) and an analog-in. Every time I switch to the analog-in I make about a dozen test reading separated by 100ms. Every time the reading start high and drop. Not stabilizing for 3 or 4 reading. That's 300ms to 400ms of waiting! I scoped the processor pin and looked at many events. The voltage during the analog-in phase is rock-steady-flat. That leaves the processor.

So, what's going on inside the processor? Why does the processor's ADC feature take hundreds of milliseconds to stabilize?